Question

Find the component form and magnitude of the vector \(u\) with the given initial and terminal points. Then find a unit vector in the direction of \(\mathbf{u}\). \(\frac{\text { Initial Point }}{(-4,3,1)}\) \(\frac{\text { Terminal Point }}{(-5,3,0)}\)

Short answer

The component form of vector \(u\) is \(-1, 0, -1\), its magnitude is \(\sqrt{2}\) and the unit vector in the direction of \(u\) is \(-\sqrt{2}/2, 0, -\sqrt{2}/2\).

Step-by-step solution

Calculate the Component Form of the Vector

Start by determining the component form of the vector \(u\). This can be done by subtracting the coordinates of the initial point from the respective coordinates of the terminal point. Thus, \(u1 = -5 - (-4) = -1\), \(u2 = 3 -3 = 0\) and \(u3 = 0 - 1 = -1\). So, the component form of the vector \(u\) is \(-1, 0, -1\).

Find the Magnitude of the Vector

Once the component form is determined, calculate the magnitude of the vector \(u\) which is the square root of the sum of the squares of the components. \(|u| = \sqrt{(-1)^2 + 0^2 + (-1)^2} = \sqrt{2}\).

Find the Unit Vector in the Direction of \(u\)

A unit vector in the direction of a given vector can be found by dividing the vector by its magnitude. In our case, we divide each component of the vector \(u\) by its magnitude \(\sqrt{2}\), so the unit vector is \(-1/\sqrt{2}, 0, -1/\sqrt{2}\) or \(-\sqrt{2}/2, 0, -\sqrt{2}/2\).

Key concepts

Component Form of Vector

Understanding the component form of a vector is foundational in vector analysis. Begin by picturing a vector as an arrow with a direction and length that represents a quantity with both magnitude and direction. In a three-dimensional setting such as the one in our exercise, vectors are expressed in terms of their x, y, and z components, like so: \[ \mathbf{v} = (v_x, v_y, v_z) \] For the provided exercise, the vector \( u \) is determined by the difference between the terminal and initial point coordinates, resulting in the component form \( -1, 0, -1 \). This subtraction essentially captures how far you need to move along each axis to get from the start to the end of the vector, which directly translates to the vector's direction and length in each dimension.

Magnitude of Vector

The magnitude of a vector, often thought of as the vector's 'length', provides us with a scalar quantity (a single number) that describes how long the vector is. It is a measure of the vector's extent in space and is irrespective of its direction. To calculate the magnitude, we use the Pythagorean theorem extended into three dimensions, summing the squares of the vector's components and then taking the square root of that sum.For example, the magnitude of vector \( u \), with components \( -1, 0, -1 \), is computed using the formula: \[ |u| = \sqrt{(-1)^2 + 0^2 + (-1)^2} = \sqrt{2} \] Knowing the magnitude is crucial for many applications, including physics problems, finding the distance between points, or normalizing vectors to create unit vectors.

Unit Vector

A unit vector is a vector whose magnitude is 1. It serves as the purest expression of direction, free from the influence of length. To find a unit vector, you divide each of the vector's components by the vector's magnitude. Essentially, you're scaling the vector down to a 'unit' length while maintaining its direction.In the exercise solution, after finding the magnitude \( \sqrt{2} \), we divide each component of vector \( u \) by \( \sqrt{2} \) to get the unit vector: \[ \frac{\mathbf{u}}{|u|} = \left( \frac{-1}{\sqrt{2}}, 0, \frac{-1}{\sqrt{2}} \right) \] Keep in mind, unit vectors are incredibly versatile tools in all fields involving vectors, including computer graphics, physics, and mathematics, as they offer a standardized way of specifying directions.

Vector Operations

Vector operations include a variety of procedures such as adding and subtracting vectors, and scalar multiplication. Addition or subtraction happens component-wise: to add or subtract two vectors, simply add or subtract their corresponding components. Scalar multiplication involves multiplying each component of a vector by a scalar (a single real number), effectively changing the vector's magnitude without affecting its direction.

For example, if we scale vector \( u \) by 2, its new components become \( 2(-1), 2(0), 2(-1) \) or \( -2, 0, -2 \). These operations are essential for translating physical ideas into mathematical form and facilitating the manipulation of vectors in various applications such as engineering, navigation, and physics.